Sketch the graph of $C$ against $x$, labelling the maximum point and the $x$-intercepts with their coordinates.

Sketch the graph of $x$ against $t$, labelling the maximum point of the graph with its coordinates.

Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.

Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph.

Use the symmetry of the graph to show that the second solution is $x=4$.

Given that $\underset{T\to \infty }{\lim }k=A$ and $\underset{T\to 0}{\lim }k=0$, sketch the graph of $k$ against $T$.

Use your graphic display calculator to find the equation of the tangent to the graph of y = f (x) at the point (–2, 38.75).

Give your answer in the form y = mx + c.

The wind speed increases. The blades rotate at twice the speed, but still at a constant rate.

At any given instant, find the probability that Tim can see point $\text{C}$ from his window. Justify your answer.

Find the time, in seconds, that point $\text{C}$ is above a height of $100 \text{m}$, during each complete rotation.

Write down the amplitude of the function.

Find the height of $\text{C}$ above the ground when $t=2$.

Sketch the function $h\left(t\right)$ for $0\le t\le 5$, clearly labelling the coordinates of the maximum and minimum points.

Write down the value of $f(1)$.

Use your graphic display calculator to find the zero of f (x).

Find the equation of the tangent line to the graph of $y=f\left(x\right)$ at $x=2$. Give the equation in the form $ax+by+d=0$ where, $a$, $b$, and $d\in \mathbb{Z}$.

Find the value of k.

Given that y = 2x³ − 9x² + 12x + 2 = k has three solutions, find the possible values of k.

Sketch the curve for −1 < x < 3 and −2 < y < 12.

Find $f^{\prime }(x)$.

A teacher asks her students to make some observations about the curve.

Three students responded.
Nadia said “The x-intercept of the curve is between −1 and zero”.
Rick said “The curve is decreasing when x < 1 ”.
Paula said “The gradient of the curve is less than zero between x = 1 and x = 2 ”.

State the name of the student who made an incorrect observation.

Using your value of k , find f ′(x).

Write down the equation for the axis of symmetry of the graph.

Write down the $x$-intercepts of the graph.

Use your graphic display calculator to find the coordinates of the local minimum point.

Draw the tangent $T$ on your graph.

Expand the expression for $f(x)$.

Write down the coordinates of the point of intersection.

On graph paper, draw the graph of $y=f\left(x\right)$ for  $-10\le x\le 4$  and  $-14\le y\le 14$. Use a scale of $1 \text{cm}$ to represent $1$ unit on the $x$-axis and $1 \text{cm}$ to represent $2$ units on the $y$-axis.

Write down the equation of $T$.

Write down the domain of the function.

Use your answer to part (b)(ii) to find the values of $x$ for which $f$ is increasing.

Find $\frac{\text{dy}}{\text{dx}}$.

Find the exact value of each of the zeros of $f$.

Use your answer to part (b) to show that the minimum value of f(x) is −22 .

Find $f^{\prime }\left(x\right)$.

Find the equation of $N$. Give your answer in the form $ax+by+d=0$ where $a$, $b$, $d\in \mathbb{Z}$.

Draw the line $y=-6$ on the axes.

An orchestra has a sound intensity of 6.4 × 10⁻³W m⁻² . Calculate the intensity level, $L$ of the orchestra.

Draw the graph of $f$ for $-3⩽x⩽3$ and $-40⩽y⩽20$. Use a scale of 2 cm to represent 1 unit on the $x$-axis and 1 cm to represent 5 units on the $y$-axis.

A rock concert has an intensity level of 112 dB. Find the sound intensity, $S$.

Find the value of $c$.

Given $f\left(a\right)=5.5$ and $f\text{'}\left(a\right)=-6$, state whether the function, $f$, is increasing or decreasing at $x=a$. Give a reason for your answer.

Find the range of values of $k$ for which $f(x)=k$ has no solution.

Write down the number of solutions to $f(x)=-6$.

Sketch the graph of $V=300\sqrt{3}x-\frac{9}{4}{x}^3$, for $0\le x\le 16$.

Find the gradient of the graph of $y=f\left(x\right)$ at $x=2$.

minimum value of $h$.

Find the time, in seconds, it takes for the blade $\text{[BC]}$ to make one complete rotation under these conditions.

maximum value of $h$.

minimum value of $h$.

Find the time, in seconds, it takes for the blade $\text{[BC]}$ to make one complete rotation under these conditions.

Calculate the angle, in degrees, that the blade $\text{[BC]}$ turns through in one second.

Find the period of the function.

Sketch the function $h\left(t\right)$ for $0\le t\le 5$, clearly labelling the coordinates of the maximum and minimum points.

At any given instant, find the probability that point $\text{C}$ is visible from Tim’s window.

maximum value of $h$.

Calculate the angle, in degrees, that the blade $\text{[BC]}$ turns through in one second.

Write down the amplitude of the function.

Find the period of the function.

Find the height of $\text{C}$ above the ground when $t=2$.

Find the time, in seconds, that point $\text{C}$ is above a height of $100 \text{m}$, during each complete rotation.

The wind speed increases and the blades rotate faster, but still at a constant rate.

Given that point $\text{C}$ is now higher than $110 \text{m}$ for $1$ second during each complete rotation, find the time for one complete rotation.